# Random Walk Simulator: √t Scaling and the Central Limit Theorem

Simulate coin-flip random walks, rescale them towards Brownian motion, see why spread grows like √t and watch the central limit theorem appear.

## About this tool

The random walk is the discrete starting point of stochastic models: add up independent ±1 coin-flip steps and watch the sum wander. This lab simulates seeded walks, rescales them by √n towards Brownian motion, measures how the spread grows with time and shows the central limit theorem turning sums of any steps into a normal distribution.

## Frequently asked questions

### What is a symmetric random walk?

A sum of independent steps that are +1 or −1 with equal probability. Its expected value stays 0, but its variance grows by 1 every step, so after n steps the typical distance from the start is √n.

### Why does the spread grow like √t and not like t?

Independent steps partly cancel. Variances of independent steps add, so variance grows linearly in time and the standard deviation, its square root, grows like √t.

### How does a random walk become Brownian motion?

Shrink the step size to 1/√n while taking n steps per unit of time. As n grows, the rescaled walk converges to Brownian motion (Donsker's principle); the Brownian Motion lab picks up from there.

### What does the central limit theorem say?

The standardised sum of many independent steps with finite variance is approximately normal, whatever the distribution of a single step. That is why the endpoint of a long walk looks Gaussian.

## Learn the theory

- [Random walks and Brownian motion](https://quantsandbox.dev/guide/stochastic/random-walks): The coin-flip walk, variance that grows linearly in time, and its limit: Brownian motion.
- [The law of large numbers and the central limit theorem](https://quantsandbox.dev/guide/stochastic/law-of-large-numbers-clt): Why averages settle down, how fast, and why the normal distribution appears: the two theorems behind Monte Carlo and the Brownian limit.
- [Probability spaces, σ-algebras and random variables](https://quantsandbox.dev/guide/stochastic/probability-spaces): Sample space, σ-algebra, probability measure, measurable maps and the σ-algebra generated by a variable.

Open the interactive tool: https://quantsandbox.dev/random-walk